1. Introduction
The notion of isotropic submanifolds of a Riemannian manifold was first introduced by B. O’Neill [
1] who studied submanifolds for which the second fundamental form is isotropic. This notion has recently be extended by Cabrerizo, Fernández and Gómez in [
2] for pseudo-Riemannian manifolds. Such manifolds are called pseudo-isotropic. Pseudo-isotropic Lagrangian surfaces has been studied in [
3]. This notion of isotropy can be extended to all bundle valued tensor fields
by saying that
is isotropic if and only if the value of
is independent of the unit vector
V. In the affine case, hypersurfaces with isotropic difference tensor
K have been studied in [
4,
5,
6].
In this paper, we study centro-affine Lorentzian surfaces
in
. The basic notions of centro-affine geometry are recalled in
Section 2. A remarkable fact about affine hypersurfaces is that we can both introduce an affine induced connection ∇ and a Levi–Civita connection
of the affine metric
h. In general those two connections do not coincide. The geometry of the hypersurface is determined by the difference tensor
K defined by
.
We say that
is pseudo-istropic if for any point
p and tangent vector
V at
p we have
where
is a function on
.
Since
is equipped with a Lorentzian metric, we can consider lightlike, timelike and spacelike vectors. We say that
is timelike (resp. spacelike) pseudo-isotropic if the Equation (
1) is satisfied for any point
p and any timelike (resp. spacelike) vector
V at
p. We say that
is lightlike pseudo-isotropic if for any point
p and lightlike vector
V at
p,
is a lightlike vector.
Note that some results in ([
2]) remain valid if we replace the second fundamental form by
K. So the following conditions are equivalent
We first show that
Theorem 1. Let be a centro-affine Lorentzian surface in . Then is pseudo-isotropic if and only if for all tangent vector field X.
Then, in
Section 4, we prove the following theorem
Theorem 2. Let be a centro-affine Lorentzian surface in , which is lightlike pseudo-isotropic but not pseudo-isotropic. Then is locally congruent with one of the following immersions
- 1.
;
- 2.
;
- 3.
;
- 4.
;
- 5.
;
- 6.
.
where , and and r is a strictly positive constant.
2. Preliminary
Here we first recall the basic notions of affine differentiable geometry and centro-affine hypersurfaces. For more details we refer to [
7,
8]. Let
be an immersion of a connected differentiable
n-dimensional manifold into the affine space
equipped with its usual flat connection
D and let
be an arbitrary local transversal vector field to
.
For any vector fields
we have
where ∇ is a torsion-free connection of
called the affine induced connection,
h a non degenerate
-tensor field called the affine metric and
S a (1-1)-tensor field called the affine shape operator.
The hypersurface is said to be a centro-affine hypersurface if the position vector is nowhere tangent to . Then f is a transversal field along itself. We can take and the Equation (3) becomes .
The following fundamental equations of Gauss and Codazzi are given by
where
X,
Y and
Z are vector fields on
.
We denote by
the Levi–Civita connection of the centro-affine metric
h and by
the curvature tensor of
. The difference tensor
K is defined by
for vector fields
X and
Y on
. We also write
and
. Thus, for each
X, it follows that
is a tensor of type
that maps
Y to
. Since both ∇ and
have zero torsion,
K is symmetric in
X and
YFrom (5) it follows that is symmetric in X, Y and Z.
On the other hand expressing the Gauss equation in terms of the Levi–Civita connection of the centro-affine metric, we deduce that
We also obtain a Codazzi equation for
KThe Tchebychev form T is defined by
3. Equiaffine Sphere and Pseudo-Isotropy
Now we suppose that
is a Lorentzian centro-affine pseudo-isotropic surface in
. The surface is equipped with an affine indefinite metric
h. Since
is pseudo-isotropic, as in the paper ([
5]), by linearization of (
1), we obtain at any point
p on
Let
, we say that
is a null frame at
p if it satisfies
,
. From (
1), we find immediately that
so
and
are lightlike vectors.
Using (
9) with
and
(resp.
and
), we get that
is orthogonal to
(resp.
).
We can write
. Using the symmetry of
in
X,
Y and
Z and the fact that
is a null frame, we prove that
Since
and
are lightlike vectors, we obtain
and
. Using the fact that
is orthogonal to
and
, we get
Therefore, we have so and . We have proved that . For any vector X at a point p, we show that so .
Conversely, suppose that
is a Lorentzian centro-affine surface in
with
. As before, we have
From , we deduce that .
Let
X a vector at a point
p, we can write
and computations give
So . Since and are independent of the basis, it is the same for . Therefore is pseudo-isotropic. This completes the proof of Theorem 1.
4. Lightlike Pseudo-Isotropic Lorentzian Surfaces
Now we suppose that is proper lightlike pseudo-isotropic but not pseudo-isotropic. We say that is lightlike pseudo-isotropic of Type 1 at the point p if there exists a lightlike vector V such that and V are independent and of Type 2 if for every lightlike vector V, and V are dependent. The points of Type 1 form an open set of and the points of Type 2 a closed set . It is clear that the null frame can be extended differentially to a neighborhood of a point p belonging either to or the interior of . Given the structure of surfaces of this section, it follows that the classification theorem remain valid on the whole of .
Lemma 1. There are no lightlike pseudo-isotropic surfaces of Type 1.
Proof. We consider a null frame in a neighborhood of
p constructed as before. We take
the lightlike vector such that
and
are independent. Then
is a multiple of
and we can choose the frame such that
. We write
. Using the symmetry of
and the fact that
is a null frame, we get
. As
is lightlike, we must have that
is also lightlike and we obtain
. From
, we get
so
. Therefore, we have
As
is the Levi–Civita connection, we have
for all vector fields on
. Using the previous expression with
and
, we prove that
and
where
and
are functions. Using the same expression, by replacing
with
or
, we get
and
. We have
Then, from the Codazzi equation (
8), we find that
and we see that
satisfies the following system of differential equations
and
.
If we compute
in two different ways, we have
and
Therefore .
It is well known that
, so
Hence, we obtain a contradiction. □
Now we deal with lightlike pseudo-isotropic surface of Type 2. We can choose a null frame such that and . The symmetry of gives . As before, we have , , and where and are functions.
Lemma 2. We have and the functions α and δ satisfy the following system of differential equations Proof. Then, from the Codazzi equation (
8), we get
and we see that
satisfies the following system of differential equations
and
.
If we compute
in two different ways, we have
and
Therefore .
From
, we get
Therefore and . □
It is easy to check that . Therefore, we have the
Corollary 1. There exist a constant and a non negative constant r such that
.
Lemma 3. There exist local coordinates u and v such that Proof. If we define vector fields
U and
V by
and
, we obtain
□
It is easy to check that the previous systems of differential equations in terms of the coordinates
reduce to
If we denote the immersion of
in
by
, using (
2), we get
We have the
Lemma 4. The immersion is determined by the following system of differential equations Proof. Recall that , and .
From the first equation, in all cases, we deduce that there exist vector valued functions and such that .
4.1. Case c = 0
Using the Corollary 1 and the derivatives of and in the direction of u and v, we get and or and .
4.1.1. Case and
The equations of the Lemma 4 become
We write .
The second equation gives
so
Then there exists a constant vector such that .
So
and finally we get
Then there exist constants vectors C, and such that and using the previous equation, we get .
We can use an affine transformation which maps the basis
to the standard basis. Then
where
,
and
.
4.1.2. Case and
The equations of the Lemma 4 become
As in the previous case, we write
. The second equation gives
Furthermore, after straightforward computations, we find that the third equation reduces to
So there exist constant vectors
and
such that
.
Moreover, there exist constant vectors and such that and by using , we show that and
Then
and
. By using an affine transformation which maps the basis
to the standard basis, we have
where
,
and
.
4.2. Case c = 1
We can suppose that
and
. The equations of the Lemma 4 become
We write .
The second equation gives
Furthermore, after straightforward computations, we find that the third equation reduces to
. Therefore we get the relation
We have
and we derive two times. Since
, we get
. Then computations give
4.2.1. Case
Then there exist constants vectors
and
such that
and using the relation
, we get
By using an affine transformation which maps the basis
to the standard basis, we have
where
,
and
.
4.2.2. Case
Then there exist constants vectors
and
such that
and using the relation
, we get
By using an affine transformation which maps the basis
to the standard basis, we have
where
,
and
.
4.2.3. Case
We have
. So there exist constant vectors
and
such that
and using the relation
, we get
By using an affine transformation which maps the basis
to the standard basis, we have
where
,
and
.
4.3. Case c = −1
We can suppose that
and
. The equations of the Lemma 4 become
As before, .
The second equation gives
Furthermore, after straightforward computations, we find that the third equation reduces to
. Therefore we get the relation
We have
and we derive two times. Since
, we have
. Then computations give
Hence there exist constants vectors
and
such that
and using the relation
, we get
By using an affine transformation which maps the basis
to the standard basis, we have
where
,
and
.
We have proved Theorem 2.